The Ameyalli-Rule: Logical Universality in a 2D Cellular Automaton

The Ameyalli-rule¹¹1Ameyalli means a spring of running water in Nahuatl, a language spoken in central Mexico. is a new result that continues our search for 2D Cellular Automata (CA) with glider-guns and eaters capable of logical universality. Since the publication of Conway’s Game-of-Life[5] with its survival/birth s23/b3 logic, many other “Life-Like” survival/birth combinations have been examined[3] but none seem to have come close[11] to achieving the complexity of behaviour of the Game-of-Life itself.

To study the basic principles of CA universal computation in a more general context, glider-guns and logical gates have been demonstrated outside survival/birth “Life-Like” constraints, but still within isotropic rule-space where all possible flips/spins of a neghborhood pattern give the same input. Isotropic rules are preferable to enact logical universality because their dynamics have no directional bias and computational machinery operates in any orientation. Examples include the 3-value 7-neighbour hexagonal Spiral-rule[1], and for a binary 2d Moore neigborhood, the Sapin-rule[13], and four rules in our own published results demonstrating logical universality — the first was the anisotropic X-Rule[7], followed by the isotropic Precursor-rule[8], the Sayab-rule[9] and the Variant-rule[10].

These rules were found from a short-list[7, 8] within an input-entropy scatter-plot[17, 18] sample of 93000+ isotropic rules, which classify rule-space between order, chaos and complexity. The input-entropy criteria in this sample follow “Life-Like” constraints to the extent that the rules are binary, isotropic, with a 3$\times$3 Moore neighborhood — — and with the $\lambda$ parameter, the density of 1s in the full 512 rule-table, similar to the Game-of-Life where $\lambda=0.273$. The short-list consists of rules that feature emergent gliders within the ordered zone of the plot. The Ameyalli-rule was found from the same short-list. Its 2-phase orthogonal emergent glider — one type found to date — is shown in figure 2. Its glider-gun (figures 3 and 4) also emerges spontaneously from a random initial state in a similar way to the Sapin-rule[13], the Sayab-rule[9], and the Spiral-rule[1], but with a lower probability, whereas the glider-guns for the other rules listed above, including the Game-of-Life, require careful construction.

The Ameyalli-rule is most efficiently defined and mutated as a 102-bit iso-rule[19], where any mutation conserves isotropy. An in-depth definition of isotropic rules, iso-rules based on iso-groups, including alternative notations — the full rule-table, symmetry classes, iso-rule prototypes, and the iso-rule in hexadecimal — are presented in section 5. These notations relate to DDLab[20] and can be redefined for Golly[6]. Figure 1 shows the Ameyalli iso-rule, and the mutations that were made to create the eaters A and B.

It turn out that a large proportion, 60/102, of Ameyalli iso-rule inputs are neutral with respect its glider-gun/eater system (section 6.1) so this system can be equivalently generated by a vast family of mutant iso-rules. The identities of the Ameyalli and other logically universal rules are centered on their glider-gun/eater systems, which is best represented by a stripped-down “idealized” version of the iso-rule with all neutral inputs set to zero.

The paper is organised into the following sections: (2) describes the glider-gun, gliders, eaters, and collisions, (3) outlines logical universality, 4 demonstrates the logical gates, (5) defines the iso-rule, (6) gives methods for iso-rule activity by the input-frequency histogram (IFH), for filtering and mutation, and for idealising the iso-rule, and 7 is a summary and discussion of the issues.

The essential ingredients for a recipe to create CA logical universality are gliders, glider-guns, eaters, and appropriate collisions.

A glider is a special kind of oscillator, a mobile pattern that recovers its form but in a displaced position, thus moving at a given velocity. A rule with the ability to support a glider, together with a stable eater, and a diversity of interactions between gliders and eaters, provides the first hint of potential universality.

Although many rules can be found with these properties, the really essential and most elusive ingredient is a glider-gun, a dynamic structure that ejects gliders periodically into space. A glider-gun can also be seen as an oscillator that adds to its form periodically to shed gliders. If can be regarded as a periodic attractor[17], especially if confined by eaters as in figure 3.

In the case of the Ameyalli-rule, as already noted, a glider-gun may emerge spontaneously from a random pattern — only one type has been observed so far. In other cases a glider-gun can sometimes be constructed from smaller components, as for the very first glider-gun created by Gosper for the Game-of-Life. These glider-guns are such elaborate structures that the probability of their spontaneous emergence is negligible.

Another essential requirement is that gliders colliding sideways can self-destruct leaving no residue, and that the resulting gap in the glider-gun-stream is sufficiently wide to allow a following glider to pass through. The Ameyalli-rule satisfies all these requirements as illustrated in figures 4 to 9.

Traditionally the proof for universality in CA is based on the Turing Machine or an equivalent mechanism, but in another approach by Conway[2], a CA is universal in the full sense if it is capable of the following,

1. 1.

   Data storage or memory.

2. 2.

   Data transmission requiring wires and an internal clock.

3. 3.

   Data processing requiring a universal set of logic gates NOT, AND, and OR, to satisfy negation, conjunction and disjunction.

This paper is confined to proving condition 3 only — for universality in the logical sense. To demonstrate universality in the full sense as for the Game-of-Life, it would be necessary to also prove conditions 1 and 2, or to prove universality in terms of the Turing Machine, as was done by Randall[12] for the Game-of-Life.

Logical universality in the Ameyalli-rule, as in the Game-of-Life, is based on Post’s Functional Completeness Theorem (FCT)[4]. This theorem guarantees that it is possible to construct a conjunctive (or disjunctive) normal form formula using only the logical gates NOT, AND and OR.

Using a specific right-angle collision, two gliders can self-destruct leaving no residue as shown in figure 6. Applying this property between glider-gun streams and a glider/gap sequence with the correct spacing and phases representing a “string” of information, its possible to build the logical gates NOT, AND and OR, illustrated in figures 7, 8 and 9. Gaps in a string are indicated by grey circles, dynamic trails are included, and B-type eaters are positioned to eventually stop gliders. Note that the leading bit of a moving glider/gap information sequence appears on the left in its string representation.

An isotropic CA rule based on a 2d binary 3$\times$3 Moore neighborhood — — can be defined by a series of methods that become ever simpler, clearer, and more concise, illustrated in figures 10, 11, and 12. In all these methods, a descending order (from left to right) of the neighborhood’s decimal equivalent is employed, in line with Wolfram’s classic convention[14, 15] — rule-tables can then be expressed in decimal or hexadecimal[18].

The decimal equivalent of a 3$\times 3$ pattern is taken as a string in the order 876
543
210 — for example, the pattern is the binary string 001110111 (119 in decimal), representing the full rule-table index 119 (figure 10), the symmetry class 119 (figure 11), and iso-rule index 66 (figure 12).

The input frequency histogram (IFH)[17, 19], a method in DDLab[20, 18], allows a dynamic view of the activity of a rule-table in relation to the current iterating space-time pattern, with options to interactively filter and mutate the current rule. Any isolated periodic pattern can be investigated to ascertain which rule-table inputs³³3The IFH applies to any type of rule-table, full and totalistic as well as iso-rules. are responsible for maintaining the pattern and to what extent.

For example, in figure 13 the space-time pattern consists of four non-stop Ameyalli gliders on a lattice with periodic boundaries. The IFH accommodates 102 columns to indicate the level of activity⁴⁴4In figures 13,14,15, the measures are averaged over a moving window of 100 time-steps to stabilise the IFH. The number of trailing hits $h_{i}$ (lookups at index $i$ of the iso-rule-table) at each time-step is recorded, and converted to $f_{i}$, the fraction (0 to 1) of all possible hits $h_{i}$$/$$(n\times w\times S)$, where $n$ is lattice size, $w$ (100) is the size of the moving window of time-steps, and $S$ (102) is the iso-rule size. $f_{i}$ is converted to log₂ to amplify infrequent hits, which sets the column height. of each iso-group — the frequency of iso-rule inputs, indexed from 101 to 0 (left to right) as in figure 12. Only 15 bars appear showing which inputs are active and to what extent, noting that the bar height is log₂ of the actual frequency to amplify infrequent hits. Mutating the iso-rule at any of these bar indices disrupts the gliders, but mutating at any other index makes no difference to the gliders themselves, but would very probably disrupt any other Ameyalli pattern. We call such inactive inputs “neutral” relative to the particular space-time pattern.

In figure 14 the (periodic) space-time pattern is the Ameyalli glider-gun system contained by eaters-B, similar to figure 3, In this case 32 iso-groups are active, the rest are neutral indicated by missing columns. In DDLab, on-the-fly key-hits will progressively filter active columns from high to low, marked by black blocks. Filtering excludes drawing these cells in the space-time pattern, or alternatively just marks the neutral bars. Other key-hits progressively mutate the iso-rule at column positions from low to high, marked by black blocks, so neutral inputs are preferentially mutated, either by flipping the input, or setting zero. In this way its possible to mutate any or all neutral inputs, and as expected, experiment confirms there is no effect on the given glider-gun system.

The Ameyalli iso-rule is another example of the search for glider-guns, then building logical gates. Its spontaneously emergent glider-gun was found from the input-entropy scatter-plot samples that favoured both order and emergent gliders. Minor mutations created two types of eater to stop the glider stream. With these ingredients, and by adjusting collision dynamics according to precise timing and points of impact, we were able to build the logical gates NOT, AND and OR required for logical universality.

Further mutations would possibly uncover other artefacts of interest, but for a better appreciation of the causal links between these significant dynamical patterns and the responsible iso-groups inputs, we applied the input-frequency histogram (IFH) method, which also revealed neutral inputs where mutations have no effect. As we have shown for the Ameyalli, setting neutral inputs to zero relative to an intensive glider-gun/eater system will reduce logically universal iso-rules to their stripped down or “idealized” form where gliders, eaters, glider-guns, and logical gates continue to be supported. The essential identity of Ameyalli, and other logically universal CA, are these dynamical objects together with the significant part of the iso-rule table that drives them. However, the possibility is there to configure neutral inputs to create other unpredicted but relevant structures.

An idealized logically universal iso-rule is the primitive of its huge family of mutants that perform the same basic functions. The dynamics of idealized versions of all the logically universal iso-rules mentioned in this paper, including the game-of-Life, are worth investigating in further work.

The experiments were realised using DDLab, Mathematica and Golly. Figures were made with DDLab. This collaborative work began in 2017 at a workshop in Ariege, France, and also at the Autonomous University of Zacatecas, Mexico and in London, UK. J. M. Gómez Soto acknowledges his residency at the DDLab Complex Systems Institute and financial support from the Research Council of México (CONACyT).